材料学基础5.pptVIP

材料科学基础Fundamental of Materials §2.5 Indices of crystal planes and directions Ⅰ.What’s crystal planes and directions? 1. Steps to determinate the plane indices: ? Establish a set of coordinate axes ? Find the intercepts of the planes to be indexed on a, b and c axes (x, y, z). ? Take the reciprocals of the intercepts 1/x, 1/y, 1/z. ? Clear fractions but do not reduce to lowest integers. ? Enclose them in parentheses, (h k l) Example: 1/2,1,2/3 2,1,3/2 (423) Plane indices referred to three axes a, b and c are also called Miller Indices. 2. The important planes in cubic crystals 3. A family of planes consists of equivalent planes so far as the atom arrangement is concerned. Ⅲ. Direction Indices 1. Derivation for the crystallographic direction As the first above, set the origin on the direction to be indexed. Find the coordinates of another point on the direction in questions. Reduce to three smallest integers: u, v, w. Enclose in square brackets [u v w]. 2. The important direction in cubic crystals: 100 : crystal axes 110 : face diagonal 111 : body diagonal 112 : apices to opposite face-centers 3. Family of directions consists of crystallographically equivalent directions, denoted u v w e.g. Ⅰ. Why choose four-axis system? Four indices has been devised for hexagonal unit cells because of the unique symmetry of the system. Ⅱ. Plane indices (hkil) It can be proved: i ≡－(h＋k) Important planes : Ⅲ. Direction indices [ u v t w ] To make the indices unique, an additional condition is imposed. ---- Let t＝－(u＋v) Transformation of indices Transformation of 3 to 4 indices, or vice versa. Suppose we have a vector, whose 3 indices [u v w], and 4 indices [u v t w]. or: For example: Examples and Discussions 1. Quick way for indexing the directions in cubic crystals: The value of a direction depends on its feature while the sign on direction. 2. The coordinate origin can